On the product of the non-linear Diamond operators related to the elastic wave

摘要

In this paper we study the solution of the equation ♢c1k♢c2ku(x)=f(x,Δc1k−1□c1k♢c2ku(x)) where ♢c1k♢c2k is the product of the Diamond operators defined by♢c1k=1c14∑i=1p∂2∂xi22−∑j=p+1p+q∂2∂xj22kand♢c2k=1c24∑i=1p∂2∂xi22−∑j=p+1p+q∂2∂xj22k,where c1 and c2 are positive constants, k is a non-negative integer, p+q=n is the dimension of the n-dimensional Euclidean space Rn, x=(x1,…,xn)∈Rn, u(x) is an unknown and f(x,Δc1k−1□c1k♢c2k) is a given function. It is found that, the existence of the solution u(x) of such equation depending on the conditions of f and Δc1k−1□c1k♢c2ku(x). Moreover such solution u(x) related to the elastic wave equation depending on the conditions of p, q and k.